**1. Introduction**

The conventional fluids, such as mixtures of ethylene glycol, oil, and water, have been used for the purpose of heat transportation by the research community. The heat transfer process was made very slow by the use of these fluids due to their poor thermal conductivity. The utilization of nanofluids as a cooling source increases operating and manufacturing costs. So, nanofluids are being used to speed up the heat transfer performances because of their excellent thermal conductivity. Nanofluids result from the suspension of submicron solid particles (nanoparticles) in the base fluids, such as water or any organic solvent. Nanoparticles are of growing interest as they play an effective role to strengthen the thermal conductivity of the base fluid. The inclusion of a magnetic field in the analysis of nanofluids has attracted much attention of researchers because of its growing applications in the fields of engineering, physics, and chemistry. The nanofluids which contain magnetic particles act as super-paramagnetic fluids which absorb the energy control of the flow and act as an alternating electromagnetic field. Nanofluids are employed as coolants in computer microchips and many other electronic devices which utilize micro-fluidic applications. With motivation from the above applications of magnetohydrodynamic flow, Sparrow and Cess [1] comprehensively analyzed the study of magnetohydrodynamic natural convection flow through the vertical plate by encountering both upward and downward flows with the e ffect of buoyancy forces. Potter and Riley [2] focused their attention on natural convection flow due to a heated sphere placed in static fluid by considering large values of the Grashof number. They discussed the characteristics of boundary layer flow into the plume numerically. Riley [3] considered the phenomenon of free convection flow along the surface of a sphere by maintaining higher temperature than the surroundings. He evaluated the model numerically for finite values of Grashof and Parndtl numbers. Andersson [4] studied the model of visco-elastic fluid over the stretching surface considering the e ffect of a transverse magnetic field analytically. Stephen and Eastman [5] proposed a novel type of fluid whose thermal conductivity is higher than conventional fluids and termed them as nanofluids. They concluded that such types of fluid enhance the thermal performances during the process of heat transfer. Samuel and Falade [6] investigated the stability of hydromagnetic fluid in porous media by incorporating the outcomes of variable viscosity. Their prediction for theoretical analysis was that an increase in the viscosity variation parameter creates a stability of the fluid flow. The transient form of the convective flow along the surface of a moving plate in a porous medium with uniform heat flux with the inclusion of a magnetic field has been studied by Al-Kabeir et al. [7]. Chamkha and Aly [8] presented nanofluids flow by means of free convection heat transfer over the permeable plate observing a magnetic field, transpiration parameter, heat absorption, and generation influences for main physical properties. The phenomenon of double di ffusive free convection nanofluids flow over the vertical plate was examined in [9]. Rosmilaet al. [10] studied the problem of free convection magnetohydrodynamic flow of nanofluids over a linearly stretching surface by the opting shooting technique along with the Runge–Kutta method of the fourth order. Mohammad et al. [11] analyzed the flow problem of a magnetohydrodynamic boundary layer over a vertical surface for nanofluids taking into account Newtonian heating e ffects. Gandhar and Reddy [12] predicted heat and mass transfer mechanism for moving plate held vertically embedded in porous media due to the insertion of magnetic field. The analysis on the influences of buoyancy force, magnetic field, and a stretching and shrinking sheet on the stagnation point flow of nanofluids was performed by Makinde et al. [13]. Olanrewaju and Makinde [14] discussed the problem of natural convection flow of nanofluids over a porous surface with a stagnation point in the presence of Newtonian heating effects. Chamkha et al. [15] reviewed the available material properties of nanofluids and focused on several geometries and applications. Stagnation point flow on a vertical stretching surface by imposing the slip condition was discussed by Khairy and Ishak [16]. The analysis of the nanofluids in the presence of a chemical reaction and magnetic field has been carried by Ltu and Ochsnor [17]. Another study was conducted to assess the free convection flow of nanofluids about di fferent circumferential points of a sphere and the fluid erupting from the boundary layer flow into the plume made above the sphere [18]. The characteristics of heat and fluid flow in the presence of nanofluids have been investigated by [19–25] along di fferent simple and complex geometries.

With inspiration from aforesaid research attempts, we intended to elaborate the problem of natural convective flow of magnetohydrodynamic nanofluids flow at the di fferent circumferential positions along the surface of a sphere and into the plume made above the sphere by encountering the e ffects of heat generation and absorption. It is necessary to highlight that no one has paid any attention towards such a problem before this attempt. In the subsequent sections, the mathematical formulation is performed and after suitable transformation of the modeled equations, a very accurate approximating technique known as the finite di fference method is directly employed to ge<sup>t</sup> the approximate solutions of the partial di fferential equations. By using FORTRAN as a computing tool, asymptotic and valid solutions of the governing model satisfying the given boundary conditions are calculated. Further, in this study, the di fferent trends/behaviors depending on various combinations of many influential parameters have been displayed graphically as well as in tabular form.

### **2. Statement of the Problem and Mathematical Formulation**

Consider a steady, two-dimensional, viscous, incompressible, and electrically conducting boundary layer flow of nanofluid. In this analysis, water is taken as the base fluid and heat generation effects are encountered. The physical sketch and geometry of the problem are shown in Figure 1. The sphere surface is kept at constant temperature *T* ˆ *w* and the nanoparticles volume fraction at the surface is *C* ˆ *w*. The coordinate along the surface of a sphere is *x*ˆ and *y*ˆ is taken as normal to the surface. The corresponding velocity components *u*ˆ and *v*ˆ is considered along and normal to the surface of the sphere respectively. There are three regions, namely sphere, fluid erupting from the boundary layer, and plume made above the sphere. The universal conservation equations for the current mechanism following Potter and Riley [2] take the forms given as below:

$$\frac{\partial(\mathfrak{H})}{\partial \mathfrak{k}} + \frac{\partial(\mathfrak{H})}{\partial \mathfrak{g}} = 0 \tag{1}$$

$$\hbar \frac{\partial \hat{\mathbf{n}}}{\partial \hat{\mathbf{x}}} + \hat{\boldsymbol{\sigma}} \frac{\partial \hat{\mathbf{n}}}{\partial \hat{\mathbf{y}}} = \nu \frac{\partial^2}{\partial^2 \hat{\mathbf{y}}} + \text{g} \boldsymbol{\beta} \{\hat{\mathbf{?}} - \hat{\mathbf{?}}\_{\text{co}}\} \text{Sim} \frac{\hat{\mathbf{x}}}{\hat{a}} + \text{g} \boldsymbol{\beta}\_{\text{c}} \{\hat{\mathbf{C}} - \hat{\mathbf{C}}\_{\text{co}}\} \text{Sim} \frac{\hat{\mathbf{x}}}{\hat{a}} - \frac{\sigma\_0 \beta\_0^2}{\rho} \hat{a} \tag{2}$$

$$\hbar \frac{\partial \hat{\boldsymbol{\Gamma}}}{\partial \boldsymbol{\hat{x}}} + \partial \frac{\partial \hat{\boldsymbol{\Gamma}}}{\partial \boldsymbol{\hat{y}}} = \alpha \frac{\partial^2 \hat{\boldsymbol{\Gamma}}}{\partial^2 \boldsymbol{\hat{y}}} + \tau \left\{ D\_B \frac{\partial \hat{\boldsymbol{\Gamma}}}{\partial \boldsymbol{\hat{y}}} \frac{\partial \hat{\boldsymbol{\Gamma}}}{\partial \boldsymbol{\hat{y}}} + \frac{D\_T}{\mathcal{T}\_{\rm co}} \left( \frac{\partial \hat{\boldsymbol{\Gamma}}}{\partial \boldsymbol{\hat{y}}} \right)^2 \right\} + \frac{Q\_0}{\rho \mathbf{C}\_p} \left( \boldsymbol{\hat{\Gamma}} - \boldsymbol{\hat{\Gamma}}\_{\rm co} \right) \tag{3}$$

$$\hbar \frac{\partial \hat{\mathbf{C}}}{\partial \hat{\mathbf{x}}} + \hbar \frac{\partial \hat{\mathbf{C}}}{\partial \hat{\mathbf{y}}} = D\_B \frac{\partial^2 \hat{\mathbf{C}}}{\partial^2 \hat{\mathbf{y}}} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 \hat{\mathbf{T}}}{\partial^2 \hat{\mathbf{y}}} \tag{4}$$

**Figure 1.** Coordinate System and Flow Geometry.

Subjected to the corresponding boundary conditions:

$$\begin{aligned} \hat{u} &= 0 = \,\,\,\,\hat{T} \quad \hat{T} = \,\,\hat{T}\_w \quad , \quad \hat{\mathbb{C}} = \,\,\hat{\mathbb{C}}\_w \quad \,\,\, \text{at} \qquad \hat{y} = 0 \\\hat{u} &\to 0 = \,\,\,\hat{\nu} \quad \,\,\,\hat{T} \to \hat{T}\_\infty \quad \,\,\,\,\hat{\mathbb{C}} \to \hat{\mathbb{C}}\_\infty \quad \,\,\, \text{as} \qquad \hat{y} \to \infty \end{aligned} \tag{5}$$

The symbols appeared in the above governing equations such as *g*, β, α, τ and β*<sup>c</sup>*, are termed as gravitational acceleration, thermal expansion of temperature, thermal diffusivity, nanoparticles heat capacity to base fluid ratio and solutal thermal expansion. The Brownian diffusion coefficient, heat generation coefficient, thermophoretic diffusion coefficient, and magnetic field strength are

denoted by *DB*, *Qo*, *DT*, and *Bo*, respectively. To make the above proposed model dimensionless, here, non-dimensionless variables are defined as below:

$$\begin{aligned} \mathbf{x} &= \frac{\mathbf{\hat{r}}}{a}, \; r(\mathbf{\hat{r}}) = a \sin \mathbf{\hat{r}}, \; y = G\_r^{-\frac{1}{2}} \frac{\mathbf{\hat{g}}}{y}, \; u = G\_r^{-\frac{1}{2}} \frac{a}{\nu} \hat{u}, \; v = G\_r^{-\frac{1}{4}} \frac{a}{\nu} \hat{v} \\\ \mathbf{G}\_r &= \frac{\mathbf{g} \beta \Delta \hat{T} a^3}{\nu^2}, \; G\_{\text{rc}} = \frac{\mathbf{g} \beta \epsilon \Delta \hat{T} a^3}{\nu^2}, \; r = \frac{\hat{r}}{a} \end{aligned} \tag{6}$$

where *a* is the sphere radius. By inserting Equation (6) into Equations (1)–(5), we obtain the following non-dimensional forms of the governing equations as given below:

$$\frac{\partial(ru)}{\partial \mathbf{x}} + \frac{\partial(rv)}{\partial y} = 0 \tag{7}$$

$$
\mu \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial^2 y} + \theta S \text{inx} + \varphi S \text{inx} - M u \tag{8}
$$

$$u\frac{\partial\theta}{\partial x} + v\frac{\partial\theta}{\partial y} = \frac{1}{\text{Pr}}\frac{\partial^2}{\partial^2 y} + \text{Nb}\frac{\partial\phi}{\partial y}\frac{\partial\theta}{\partial y} + \text{Nt}\left(\frac{\partial\theta}{\partial y}\right)^2 + \text{Q}\theta\tag{9}$$

$$u\frac{\partial\varphi}{\partial\mathbf{x}} + v\frac{\partial\varphi}{\partial y} = \frac{1}{\text{Sc}} \left( \frac{\partial^2}{\partial^2 y} + \frac{\text{Nt}}{\text{Nb}} \frac{\partial^2}{\partial^2 y} \right) \tag{10}$$

,

With boundary conditions:

$$\begin{aligned} u = 0 &= v & \theta &= 1, & \quad \varphi = 1 & \quad & at & \quad y = 0\\ u &\to 0, & \quad \varphi &\to 0, & \quad \theta &\to 0 & \quad & \quad y \to \infty \end{aligned} \tag{11}$$

where

$$\text{Pr} = \frac{\nu}{\alpha}, \text{ Sc} = \frac{\nu}{D\_b}, \text{ N}\_{\text{b}} = \frac{(\rho c)\_p D\_b (t \text{5}\rho\_{\text{w}} - \rho\_{\text{co}})}{(\rho c)\_f \nu}, \text{ N}\_{\text{t}} = \frac{(\rho c)\_p D\_{\text{T}} (T\_{\text{w}} - T\_{\text{co}})}{(\rho c)\_f T\_{\text{co}} \nu}$$

The parameters appearing above are the thermophoresis parameter, Schmidt number, Prandtl number, and Brownian motion parameter, which are designated Nt, Sc, Pr, and *Nb*. Here, *M* = <sup>σ</sup>0β20 ρν *a*2 *G*1/2 *r* , and Q = *Q*0 μ*Cp a*2 *G*1/2 *r* represent the magnetic field parameter and heat generation parameter, respectively.
